Finite Group Databasebeta

← #28312276

Subgroups — #28312276

Order 13568.

488 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290579C211yesmaximal, characteristic
2C4×#13133275C211yesmaximal, characteristic
3#20289800C214yesmaximal
4#20279033C211yesmaximal, characteristic
5#16811626—21no—
6C2×#13133277C2212yes—
7C2×#13133275C2212yes—
8C4×C22⋊#6157—21no—
9#16811503—21no—
10#16811501—21no—
11C2×C4×C22⋊D53C2211yescharacteristic
12C22×C4⋊Dic53C411yescharacteristic, complemented
13#16810563—21no—
14#16810560—21no—
15#16810522—21no—
16#16810500C2212yes—
17#16810498C412yescomplemented
18#16810480—21no—
19C53×#380C411yescharacteristic, complemented
20C2×#11719940—22no—
21C2×C22⋊#6157—22no—
22C2×C4×#6157—41no—
23#13133284—41no—
24C4⋊#88324—41no—
25#13133277—24no—
26#13133275—22no—
27#13133275Q814yes—
28(C2×#6157)⋊C4—41no—
29C22×C22⋊D53C2311yescharacteristic
30C23×Dic53C2×C412yes—
31C22×C4×D53D411yescharacteristic
32C4×C22⋊D53—41no—
33C4×C22⋊D53—41no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53—41no—
38C2×C4⋊Dic53—41no—
39C2×#11719919D411yescharacteristic
40C2×C22⋊Dic53C2×C411yescharacteristic
41C2×#11719921—22no—
42C2×#11719921D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44#13133124—22no—
45#13133124—22no—
46C23×C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C412yes—
49#95×C53—21no—
50C24×C53C22×C411yescharacteristic
51C23×D53C2×D411yescharacteristic
52C22×#6157—42no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—42no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C2×C22⋊D53—41no—
60C2×C22⋊D53—41no—
61C2×C22⋊D53C2×Q812yes—
62C2×C22⋊D53#4112yes—
63C22×Dic53—22no—
64C22×Dic53—42no—
65C22×Dic53—82no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—42no—
72C22⋊#6157—24no—
73C22⋊#6157—42no—
74#11719940—42no—
75#11719940—42no—
76C2×#88324—42no—
77C4×#6157—82no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C22⋊Dic53—41no—
81#11719921—41no—
82#11719921—82no—
83C4⋊Dic53—42no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86C4⋊Dic53—82no—
87#11719919—41no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53C4×Q812yes—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53C22⋊Q812yes—
101C2×#6157—44no—
102C2×#6157—82no—
103C2×#6157—82no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53#9911yescharacteristic
114C22⋊D53—42no—
115C22⋊D53—42no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—84no—
126C2×Dic53#12411yescharacteristic
127C2×Dic53C4⋊Q811yescharacteristic
128C4×D53—42no—
129C4×D53—81no—
130#88324—82no—
131#13295—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53#39111yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53#41011yescharacteristic
142C2×D53—22no—
143C2×D53—41no—
144C2×D53—41no—
145C2×D53—42no—
146C2×D53—81no—
147#6157—84no—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53—82no—
151C4×C53#35212yes—
152Dic53—21no—
153Dic53—21no—
154Dic53—41no—
155Dic53—41no—
156Dic53—82no—
157#2487—531no—
158#1800—531no—
159C4×#99—531no—
160#1390—534no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—42no—
164C2×C53#199611yescharacteristic
165C2×C53C2×#35211yescharacteristic
166C2×C53#212211yescharacteristic
167D53—42no—
168C2×C4×D4—531no—
169C22×C4⋊C4—531no—
170#423—1061no—
171C2×#100—532no—
172C2×#99—532no—
173#400—1061no—
174#399—1061no—
175#389—1061no—
176#386—532no—
177#385—532no—
178#380—1061no—
179#380#615711yescharacteristic, complemented
180C4×C22⋊C4—1061no—
181#343—1061no—
182#342—1061no—
183C53#1329511yescharacteristic, complemented, Sylow
184C22×D4—531no—
185C23×C4—531no—
186C23×C4—532no—
187C23×C4C2×#615711yescharacteristic
188C2×M16—1062no—
189C4×D4—2121no—
190C4×D4—2121no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1062no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—2121no—
197C2×C4⋊C4—2121no—
198C2×C22⋊C4—531no—
199C2×C22⋊C4—531no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4C2×#615712yes—
203C2×C42—531no—
204C2×C42—2121no—
205C4⋊C8—2121no—
206C4⋊C8—2121no—
207#100—1064no—
208#99—534no—
209#99—1062no—
210#95—21no—
211#95—1062no—
212#95—1062no—
213#95—2121no—
214C24—531no—
215C24C22×#615711yescharacteristic
216C2×D4—532no—
217C2×D4—532no—
218C2×D4—2121no—
219C2×D4—2121no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—1062no—
230C22×C4—4242no—
231C22×C4—2122no—
232C22×C4—2122no—
233C22×C4—2122no—
234C22×C4C22⋊#615711yescharacteristic
235C22×C4C22⋊#615711yescharacteristic
236M16—2122no—
237M16—2122no—
238C2×C8—2122no—
239C4⋊C4—42no—
240C4⋊C4—4242no—
241C4⋊C4—2121no—
242C4⋊C4—2122no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C22⋊C4—42no—
246C22⋊C4—1064no—
247C22⋊C4—4242no—
248C22⋊C4—2121no—
249C22⋊C4—2121no—
250C22⋊C4—2122no—
251C42—1061no—
252C42—1061no—
253C42—4242no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—532no—
259C23—1061no—
260C23—1061no—
261C23—2121no—
262C23—2121no—
263C23C2×C22⋊#615711yescharacteristic
264C23Q8×#615712yes—
265D4—2122no—
266D4—2122no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—531no—
276C2×C4—531no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—4241no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—4244no—
285C2×C4—2122no—
286C2×C4—2122no—
287C2×C4—2124no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4#1313327611yescharacteristic
293C2×C4#1313327611yescharacteristic
294C8—4242no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—1062no—
303C22—4241no—
304C22—2122no—
305C22—2121no—
306C22—2121no—
307C22(C2×#6157)⋊Q811yescharacteristic
308C22C2×#1313327611yescharacteristic
309C22#1681160311yescharacteristic
310C4—21no—
311C4—41no—
312C4—82no—
313C4—1061no—
314C4—1061no—
315C4—4244no—
316C4—4242no—
317C4—2121no—
318C4—2121no—
319C4#1681152612yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—2122no—
324C2#2029090511yescharacteristic
325C2C2×#1681152611yescharacteristic
326C2#2029097611yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC53×#380C4
socleC22×C53C2×#99
prime radical, cosocleC22×C22⋊D53C23
hypercenter#380#6157
hypocenterC53#13295
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. C22×C4⋊Dic53C4semidirect
17. #16810498C4semidirect
19. C53×#380C4semidirect
179. #380#6157semidirect
183. C53#13295semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202905792
C4×#131332752
#202898002
#202790332
#1329553